The Topological Tverberg Theorem and winding numbers

Abstract

The Topological Tverberg Theorem claims that any continuous map of a ( q - 1 ) ( d + 1 ) -simplex to R d identifies points from q disjoint faces. (This has been proved for affine maps, for d ⩽ 1 , and if q is a prime power, but not yet in general.) The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a “Winding Number Conjecture” that concerns only maps of the ( d - 1 ) -skeleton of a ( q - 1 ) ( d + 1 ) -simplex to R d . “Many Tverberg partitions” arise if and only if there are “many q-winding partitions.” The d = 2 case of the Winding Number Conjecture is a problem about drawings of the complete graphs K 3 q - 2 in the plane. We investigate graphs that are minimal with respect to the winding number condition.